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Model the room's decaying acoustic waves as a weakly damped resonator. The quality factor from ringdown is related to stored energy $E$ bywhere $A$ is the pressure wave amplitude and $\omega =2\pi f$. The measured frequency $f=440\mathrm{Hz}$ gives $\omega \simeq 2.76\times 10^3{\mathrm{s}}^{-1}$. If the stated $5\mathrm{s}$ is interpreted as an approximate wave amplitude exponential relaxation time,an estimate of order $10^4$. Taking $5\mathrm{s}$ as an energy exponential relaxation time instead gives $Q\simeq 1.4\times 10^4$.

There is a real convention ambiguity: “heard to die away” gives neither an initial sound level nor an audibility threshold, so it does not determine either exponential relaxation time. More generally, if the energy falls by a factor $E(0)/E(t_d)$ in $t_d$,In room acoustics, a reverberation time $T_{60}$ usually denotes a decibel drop of $60\mathrm{dB}$, or an energy factor $10^6$. If $5\mathrm{s}$ means $T_{60}$, the corresponding estimate is instead $\boxed{Q\simeq 2\pi (440)(5)/(6\ln 10)\simeq 1.0\times 10^3}$. The simple decay-time estimate and the conventional reverberation time estimate should not be silently equated.

Reverberation is the persistence of acoustic waves after a source stops, produced by repeated reflections in an enclosure. A diffuse-room approximation gives an approximately exponential decay of mean acoustic energy; an individual normal mode may instead be described as a damped resonator.

A sonic flow has Mach number exactly one: the local fluid speed relative to the body equals the local sound speed. In uniform gas flowing at $U=c$ past a resting body, an upstream-directed acoustic wave has zero body-frame velocity. A sonic point is a location where this equality holds. In the moving-point acoustic wavefronts construction, old spheres touch the source at the origin; the Mach angle tends to $90^{\circ }$ as the supersonic flow approaches this threshold.

A subsonic flow has Mach number less than one, so its speed is below the local sound speed.

For uniform gas moving at $U{\mathbf{e}}_x$ past a resting body, an upstream-directed acoustic wave has body-frame velocity $(U-c){\mathbf{e}}_x$. When $0\le M<1$, this velocity points upstream. This is why the body can influence the upstream fluid flow through small pressure disturbances.

A supersonic flow has Mach number greater than one, so the flow speed exceeds the local sound speed.

For uniform gas moving at $U{\mathbf{e}}_x$ past a resting body, even an acoustic wave directed against the fluid flow has body-frame velocity $(U-c){\mathbf{e}}_x$ downstream. More generally, an acoustic wave travelling along a unit vector $\mathbf{n}$ has body-frame velocity $U{\mathbf{e}}_x+c\mathbf{n}$; its $x$ component is at least $U-c>0$. All such weak signals move downstream and their possible directions fit inside a Mach cone.

An ultrasonic motion sensor estimates distance from the return delay of an acoustic wave pulse reflected by a target. The teaching setup places a flat flag on the cart facing the detector above the air track, and specifies a minimum range of about $0.3\mathrm{m}$ for its detector. That apparatus-specific limit should not be assumed for every sensor model. ^[ref]

The wave equation is a second-order linear partial differential equation describing waves with finite propagation speed. In one spatial dimension its homogeneous, constant-coefficient form isHere $x$ is position, $t$ is time, $u(x,t)$ is the disturbance, and subscripts denote partial derivatives. For a wave equation on a string, $u$ is transverse displacement; in linear acoustics, it can be a pressure perturbation. In several spatial dimensions, $u_{xx}$ becomes the Laplacian $\Delta u$. The same mathematical structure connects vibrating strings and acoustic waves, although their physical assumptions and wave speeds differ. ^[ref]

For the whole-line Cauchy problem, prescribe both displacement $u(x,0)=f(x)$ and velocity $u_t(x,0)=g(x)$. A classical solution has continuous partial derivatives through order two; $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ suffice for D'Alembert's formula to produce one. A finite string additionally needs boundary conditions. These statements concern the unforced model with constant $c$; damping, spatially varying coefficients and nonlinear terms change the solution formula.

A travelling wave $F(x-ct)$ carries its entire profile to the right at wave speed $c$; $G(x+ct)$ travels left. Their sum is again a solution by the superposition principle. A standing wave instead has a fixed mode shape whose size and sign oscillate: in $\sin (kx)\cos (ckt)$, each node of a standing wave stays stationary. The distinction is especially clear in the pulse and string diagrams below. For this ideal model, the dispersion relation is ${\omega }^2=c^2{k}^2$: each propagation branch is nondispersive, so a localized profile translates without spreading. This makes the equation a useful reference against which to understand wave dispersion and damping.

The appealing combination is local interaction with global predictability: neighbouring displacements determine acceleration through $u_{xx}$, yet D'Alembert's formula explicitly identifies exactly which initial locations can influence a later observation. Finite propagation speed turns that observation into a domain of dependence, rather than requiring knowledge of the entire initial string.

For the homogeneous one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$ with constant $c>0$, wave speed means the speed at which a travelling wave profile translates. In $F(x-ct)$, a feature initially at $x=s$ follows $x=s+ct$, giving signed velocity $+c$; in $G(x+ct)$ it follows $x=s-ct$, giving signed velocity $-c$. The characteristic lines for the one-dimensional wave equation therefore have slopes $dx/dt=\pm c$. The speed of either profile is $c$, even though the displacement at a fixed point can change sign or pass through zero.^[ref]

The same $c$ bounds the causal cone for the constant-speed wave equation. For a wave equation on a string, $c=\sqrt{T/\mu }$ depends on tension and linear mass density; for small acoustic waves in stationary uniform air it is the speed of sound. This interpretation assumes the stated ideal wave equation. In media with wave dispersion, phase velocity, group velocity and the earliest signal arrival have different meanings: a measured wave crest speed need not be the speed of a wave envelope, and neither alone establishes a causal cone for the constant-speed wave equation.